In LR, what are your Toning settings and Detail settings: the settings that add contrast and the settings that mitigate noise?

If I start with your NEF and import which gives me Adobe Factory Defaults since I haven't defined any customer default, then do Dehaze +70 to give the sky more contrast like in your image, although you probably added contrast differently, I can see quite a bit if ISO noise in the sky.

To mitigate that noise I turn up the Sharpen Mask slider to 90 so we're only sharpening edges, and I can still see a bit of ISO noise in the sky so I add some Luminance NR of +25, then I see no ISO noise in the sky and unlike your situation, my export matches the screen preview reasonably well although the change in Color Profile to sRGB during Export has made the sky slightly darker in the darkest areas.

Without knowing what settings are applied to your NEF before you exported, it's still not clear if the issue is the display of the preview that shows no noise or the Export that is adding noise, somehow, for you.

In my case if I mitigate the noise in the onscreen preview I don't see much in the exported image, either.

Here is a side-by-side before and after changing the Detail settings which demonstrates the sky noise I was seeing without some Masking and Lum NR:

Here is a side-by-side showing no noise on the preview nor the exported image, once I've mitigated the noise in the preview:

Thanks for taking the time to try out all of that. I've attached a couple of screenshots of my Develop settings. I'm using Lightroom CC 2015.10.1., dehaze is set to +30, some sharpening and noise reduction settings are used.

As I was saying before, I've made similar changes to other photos before without running into this issue. Now it's coming up in each photo I edit/try to export.